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which some of Ms contemporaries imagined it to be, and it is
admitted that there may be something - not much, perhaps -
in his disturbing objections. Whether there is or not, Kro-
neeker's attack was partly responsible for the third period of
rigour hi modern mathematical reasoning, that which we our-
selves are attempting to enjoy. Weierstrass was not the only
fellow-mathematician whom Kronecker harried; Cantor also
suffered deeply under what he considered his influential col-
league^ malicious persecution- All these men will speak for
themselves in the proper place; here we are only attempting to
indicate that their lives and work were closely interwoven in
at least one corner of the gorgeous pattern.
To complete the picture we must indicate other points of
contact between Weierstrass, Kronecker, and Riemann on one
side and Kronecker and DedeMnd on the other. Abel, we recall,
died in 1829, Galois in 1832, and Jacobi hi 1851. In the epoch
under discussion one of the outstanding problems hi mathe-
matical analysis was the completion of the work of Abel and
Jacobi on multiple periodic functions - elliptic functions,
Abelian functions (see chapters 17, 18). From totally different
points of -view Weierstrass and Riemann accomplished what
was to be done - Weierstrass indeed considered himself hi some
degree a successor of Abel; Kronecker opened up new vistas in
elliptic functions but he did not compete with the other two in
the field of Abelian functions. Kronecker was primarily an
arithmetician and an algebraist; some of his best work went
into the elaboration and extension of the work of Galois in the
theory of equations. Thus Galois found a worthy successor not
too long after his death.
Apart from his forays into the domain of continuity and irra-
tional numbers, Dedekind's most original work was hi the
higher arithmetic, which he revolutionized and. renovated. In
this Kroneeker was his able and sagacious rival, but again their
whole approaches were entirely different and characteristic of
the two men: Dedekind overcame his difficulties in the theory
of algebraic numbers by taking refuge in the infinite (in his
theory of 'ideals', as will be indicated in the proper place);
Kronecker sought to solve his problems in the finite.